package ‘gmp’package ‘gmp’ february 15, 2013 version 0.5-4 date 2013-02-04 title multiple...

Package ‘gmp’February 15, 2013

Version 0.5-4

Date 2013-02-04

Title Multiple Precision Arithmetic

Author Antoine Lucas, Immanuel Scholz, Rainer Boehme, Sylvain Jasson, Martin Maechler

Maintainer Antoine Lucas

Description Multiple Precision Arithmetic (big integers and rationals,prime number tests, matrix com-putation), ‘‘arithmetic without limitations’’ using the C library GMP (GNU Multiple PrecisionArithmetic).

Depends methods

Suggests Rmpfr

SystemRequirements gmp (>= 4.2.3)

License GPL

BuildResaveData no

LazyDataNote not available, as we use data/*.R *and* our classes

URL http://mulcyber.toulouse.inra.fr/projects/gmp

Repository CRAN

Date/Publication 2013-02-05 18:42:33

NeedsCompilation yes

1

http://mulcyber.toulouse.inra.fr/projects/gmp

2 apply

R topics documented:apply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2asNumeric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Bigq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4bigq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Bigq operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7bigz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8bigz operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11cumsum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Extract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Extremes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15factorialZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17formatN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18frexpZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19gcd.bigz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20gcdex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21gmp.utils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22is.whole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22isprime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23lucnum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27nextprime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Oakley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29powm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Random . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Relational Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32sizeinbase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32solve.bigz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Stirling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Index 37

apply Apply Functions Over Matrix Margins (Rows or Columns)

Description

These are S3 methods for apply() which we re-export as S3 generic function. They “overload”the apply() function for big rationals ("bigq") and big integers ("bigz").

Usage

## S3 method for class ’bigz’apply(X, MARGIN, FUN, ...)## S3 method for class ’bigq’apply(X, MARGIN, FUN, ...)

asNumeric 3

Arguments

X a matrix of class bigz or bigq, see e.g., matrix.bigz.

MARGIN 1: apply function to rows; 2: apply function to columns

FUN function to be applied

... (optional) extra arguments for FUN(), as e.g., in lapply.

Value

The bigz and bigq methods return a vector of class "bigz" or "bigq", respectively.

Author(s)

Antoine Lucas

See Also

apply; lapply is used by our apply() method.

Examples

x

4 Bigq

Value

an R object of type (typeof) "numeric", a matrix or array if x had non-NULL dimension dim().

Methods

signature(x = "ANY") the default method, which is the identity for numeric array.

signature(x = "bigq") the method for big rationals.

signature(x = "bigq") the method for big integers.

Note that package Rmpfr provides methods for its number-like objects.

Author(s)

Martin Maechler

See Also

as.numeric coerces to both "numeric" and to a vector, whereas asNumeric() should keep dim(and other) attributes.

Examples

m

bigq 5

e1 >= e2## S3 method for class ’bigq’e1 > e2## S3 method for class ’bigq’e1 != e2

Arguments

x, e1, e2 Object or vector of class bigq

Examples

x

6 bigq

... additional arguments passed to methods

quote (for printing:) logical indicating if the numbers should be quoted (as charactersare); the default used to be TRUE (implicitly) till 2011.

initLine (for printing:) logical indicating if an initial line (with the class and length ordimension) should be printed.

Details

as.bigz.bigq() returns the smallest integers not less than the corresponding rationals bigq.

Value

An R object of (S3) class "bigq" representing the parameter value.

Author(s)

Antoine Lucas

References

http://mulcyber.toulouse.inra.fr/projects/gmp/

Examples

x

Bigq operators 7

Bigq operators Basic arithmetic operators for large rationals

Description

Addition, subtraction, multiplication, division, and absolute value for large rationals, i.e. "bigq"class R objects.

Usage

add.bigq(e1, e2)## S3 method for class ’bigq’e1 + e2

sub.bigq(e1, e2=NULL)## S3 method for class ’bigq’e1 - e2

mul.bigq(e1, e2)## S3 method for class ’bigq’e1 * e2

div.bigq(e1, e2)## S3 method for class ’bigq’e1 / e2

## S3 method for class ’bigq’e1 ^ e2

## S3 method for class ’bigq’abs(x)

Arguments

e1,e2, x of class "bigq", or (e1 and e2) integer or string from an integer

Details

Operators can be use directly when the objects are of class "bigq": a + b, a * b, etc, and a ^ n,where n must be coercable to a biginteger ("bigz").

Value

A bigq class representing the result of the arithmetic operation.

Author(s)

Immanuel Scholz and Antoine Lucas

8 bigz

References

http://mulcyber.toulouse.inra.fr/projects/gmp/

Examples

## 1/3 + 1 = 4/3 :as.bigq(1,3) + 1

r

bigz 9

Details

Bigz’s are integers of arbitrary, but given length (means: only restricted by the host memory).Basic arithmetic operations can be performed on bigzs as addition, subtraction, multiplication, divi-sion, modulation (remainder of division), power, multiplicative inverse, calculating of the greatestcommon divisor, test whether the integer is prime and other operations needed when performingstandard cryptographic operations.

For a review of basic arithmetics, see add.bigz.

Comparison are supported, i.e., "==", "!=", "=".

Objects of class "bigz" may have an attribute mod which specifies a modulus that is applied aftereach arithmetic operation. When the object has such a modulus m, all arithmetic is performed“modulo m”, i.e.,

result

10 bigz

References

The GNU MP Library, see http://gmplib.org

Examples

## 1+1=2a

bigz operators 11

bigz operators Basic Arithmetic Operators for Large Integers ("bigz")

Description

Addition, substraction, multiplication, (integer) division, remainder of division, multiplicative in-verse, power and logarithm functions.

Usage

add.bigz(e1, e2)sub.bigz(e1, e2 = NULL)mul.bigz(e1, e2)div.bigz(e1, e2)divq.bigz(e1,e2) ## == e1 %/% e2mod.bigz(e1, e2) ## == e1 %% e2## S3 method for class ’bigz’abs(x)

inv.bigz(a, b,...)## == (1 / a) (modulo b)

pow.bigz(e1, e2,...)## == e1 ^ e2

## S3 method for class ’bigz’log(x, base=exp(1))## S3 method for class ’bigz’log2(x)## S3 method for class ’bigz’log10(x)

Arguments

x bigz, integer or string from an integer

e1, e2, a,b bigz, integer or string from an integer

base base of the logarithm; base e as default

... Additional parameters

Details

For details about the internal modulus state, see the manpage of bigz.

div or "/" return a rational number; divq or "%/%" return the quotient of integer division.

Operators can be used directly when objects are of class bigz: a + b, log(a), etc.

Value

A bigz class representing the result of the arithmetic operation.

12 cumsum

Author(s)

Immanuel Scholz and Antoine Lucas

References


Examples

# 1+1=2as.bigz(1) + 1as.bigz(2)^10as.bigz(2)^200

# if my.large.num.string is set to a number, this returns the least byte(my.large.num.string 1 (mod 11)inv.bigz(7, 11)## hence, 8a

cumsum 13

Usage

## S3 method for class ’bigz’cumsum(x)## S3 method for class ’bigq’cumsum(x)## S3 method for class ’bigz’sum(..., na.rm = FALSE)## S3 method for class ’bigq’sum(..., na.rm = FALSE)## S3 method for class ’bigz’prod(..., na.rm = FALSE)## S3 method for class ’bigq’prod(..., na.rm = FALSE)

Arguments

x, ... R objects of class bigz or bigq or ‘simple’ numbers.

na.rm logical indicating if missing values (NA) should be removed before the compu-tation.

Value

return an element of class bigz or bigq.

Author(s)

Antoine Lucas

See Also

apply

Examples

x

14 Extract

Extract Extract or Replace Parts of an Object

Description

Operators acting on vectors, arrays and lists to extract or replace subsets.

Usage

## S3 method for class ’bigz’c(..., recursive = FALSE)## S3 method for class ’bigq’c(..., recursive = FALSE)## S3 method for class ’bigz’rep(x,times, ...)## S3 method for class ’bigq’rep(x,times, ...)

Arguments

x R object of class "bigz" or "bigq", respectively.

... further arguments, notably for c().

times integer

recursive unused here

Note

Unlike standard matrices, x[i] and x[i,] do the same.

Examples

a extends the vectors (filling with NA)a[2]

Extremes 15

m[-c(2,3),]m[-c(2,3)]m[c(TRUE,FALSE,FALSE)]

##_modification on matrixm[2,-1]

16 factorialZ

Examples

x

factorization 17

system.time(replicate(8, f1e4

18 formatN

formatN Format Numbers Keeping Classes Distinguishable

Description

Format (generalized) numbers in a way that their classes are distinguishable. Contrary to format()which uses a common format for all elements of x, here, each entry is formatted individually.

Usage

formatN(x, ...)## Default S3 method:formatN(x, ...)## S3 method for class ’integer’formatN(x, ...)## S3 method for class ’double’formatN(x, ...)## S3 method for class ’bigz’formatN(x, ...)## S3 method for class ’bigq’formatN(x, ...)

Arguments

x any R object, typically “number-like”.

... potentially further arguments passed to methods.

Value

a character vector of the same length as x, each entry a representation of the corresponding entryin x.

Author(s)

Martin Maechler

See Also

format, including its (sophisticated) default method; as.character.

Examples

## Note that each class is uniquely recognizable from its output:formatN( -2:5)# integerformatN(0 + -2:5)# double precisionformatN(as.bigz(-2:5))formatN(as.bigq(-2:5, 4))

frexpZ 19

frexpZ Split Number into Fractional and Exponent of 2 Parts

Description

Breaks the number x into its binary significand (“fraction”) d ∈ [0.5, 1) and ex, the integral expo-nent for 2, such that x = d · 2ex.If x is zero, both parts (significand and exponent) are zero.

Usage

frexpZ(x)

Arguments

x integer or big integer (bigz).

Value

a list with the two components

d a numeric vector whose absolute values are either zero, or in [ 12 , 1).

exp an integer vector of the same length; note that exp == 1 + floor(log2(x)),and hence always exp > log2(x).

Author(s)

Martin Maechler

See Also

log2, etc; for bigz objects built on (the C++ equivalent of) frexp(), actually GMP’s ‘mpz_get_d_2exp()’.

Examples

frexpZ(1:10)## and confirm :with(frexpZ(1:10), d * 2^exp)x

20 gcd.bigz

gcd.bigz Greatest Common Divisor, Least Common Multiple

Description

Compute the greatest common divisor (GCD) and least common multiple (LCD) of two (big) inte-gers.

Usage

## S3 method for class ’bigz’gcd(a, b)lcm.bigz(a, b)

Arguments

a,b Either integer, numeric, bigz or a string value; if a string, either starting with 0xfor hexadecimal, 0b for binary or without prefix for decimal values.

Value

An element of class bigz

Author(s)

Antoine Lucas

References


See Also

gcdex

Examples

gcd.bigz(210,342) # or alsolcm.bigz(210,342)a

gcdex 21

gcdex Compute Bezoult coefficient

Description

Compute g,s,t as as+ bt = g = gcd(a, b). s and t are also known as Bezoult coefficients.

Usage

gcdex(a, b)

Arguments

a,b Either integer, numeric or string value (String value: ither starting with 0x forhexadecimal, 0b for binary or without prefix for decimal values.) Or an elementof class bigz.

Value

3 values:

g,s,t Elements of class bigz

Author(s)

Antoine Lucas

References


See Also

gcd.bigz

Examples

gcdex(342,654)

http://gmplib.org

22 is.whole

gmp.utils GMP Number Utilities

Description

gmpVersion() returns the version of the GMP library which gmp is currently linked to.

Usage

gmpVersion()

References


Examples

gmpVersion()

is.whole Whole ("Integer") Numbers

Description

Check which elements of x[] are integer valued aka “whole” numbers.

Usage

is.whole(x)## Default S3 method:is.whole(x)## S3 method for class ’bigz’is.whole(x)## S3 method for class ’bigq’is.whole(x)

Arguments

x any R vector

Value

logical vector of the same length as x, indicating where x[.] is integer valued.

Author(s)

Martin Maechler

http://gmplib.org

isprime 23

See Also

is.integer(x) (base package) checks for the internal mode or class; not if x[i] are integer valued.

The is.whole() method for "mpfr" numbers.

Examples

is.integer(3) # FALSE, it’s internally a doubleis.whole(3) # TRUE## integer valued complex numbers (two FALSE) :is.whole(c(7, 1 + 1i, 1.2, 3.4i, 7i))is.whole(factorialZ(20)^(10:12)) ## "bigz" are *always* whole numbersq

24 lucnum

Author(s)

Antoine Lucas

References


See Also

nextprime, factorize.

Note that for “small” n, which means something like n < 10′000′000, non-probabilistic methods(such as factorize()) are fast enough. For example, primes in package sfsmisc.

Examples

isprime(210)isprime(71)

# All primes numbers from 1 to 100t 0]

table(isprime(1:10000))# 0 and 2 : surely prime or not prime

primes

matrix 25

Description

fibnum compute n-th Fibonacci number. fibnum2 compute (n-1)-th and n-th Fibonacci number.lucnum compute n-th lucas number. lucnum2 compute (n-1)-th and n-th lucas number.

Fibonacci numbers are define by: Fn = Fn−1 + Fn−2 Lucas numbers are define by: Ln = Fn +2Fn−1

Usage

fibnum(n)fibnum2(n)lucnum(n)lucnum2(n)

Arguments

n Integer

Value

Fibonacci numbers and Lucas number.

Author(s)

Antoine Lucas

References


Examples

fibnum(10)fibnum2(10)lucnum(10)lucnum2(10)

matrix Matrix manipulation with gmp

Description

Overload of “all” standard tools useful for matrix manipulation adapted to large numbers.

http://gmplib.org

26 matrix

Usage

## S3 method for class ’bigz’matrix(data = NA, nrow = 1, ncol = 1, byrow = FALSE, dimnames = NULL, mod = NA,...)

## S3 method for class ’bigz’x %*% y## S3 method for class ’bigq’x %*% y## S3 method for class ’bigq’crossprod(x, y=NULL)## S3 method for class ’bigz’tcrossprod(x, y=NULL)## ..... etc

Arguments

data an optional data vectornrow the desired number of rowsncol the desired number of columnsbyrow logical. If FALSE (the default), the matrix is filled by columns, otherwise the

matrix is filled by rows.dimnames not implemented for "bigz" or "bigq" matrices.mod optional modulus (when data is "bigz").... Not usedx,y numeric, bigz, or bigq matrices or vectors.

Details

Extract function is the same use for vector or matrix. Then, x[i] return same value as x[i,].

Special features concerning the "bigz" class: the modulus can be

Unset: Just play with large numbersSet with a vector of size 1: Example: matrix.bigz(1:6,nrow=2,ncol=3,mod=7) This means you

work in Z/nZ, for the whole matrix. It is the only case where the %*% and solve functionswill work in Z/nZ.

Set with a vector smaller than data: Example: matrix.bigz(1:6,nrow=2,ncol=3,mod=1:5). Then,the modulus is repeated to the end of data. This can be used to define a matrix with a differentmodulus at each row.

Set with same size as data: Modulus is defined for each cell

Value

A matrix of class bigz or bigq

Author(s)

Antoine Lucas

modulus 27

See Also

Solving linear algebra system solve.bigz; matrix

Examples

V

28 nextprime

Description

The modulus of a bigz number a is “unset” when a is a regular integer, a ∈ Z). Or the moduluscan be set to m which means a ∈ Z/m · Z), i.e., all arithmetic with a is performed ‘modulo m’.

Usage

modulus(a)modulus(a)

Oakley 29

Details

This function uses probabilistic algorithm to identify primes. For practical purposes, it is adequate,the chance of a composite passing will be extremely small.

Value

A (probably) prime number

Author(s)

Antoine Lucas

References


See Also

isprime and its references and examples.

Examples

nextprime(14)## still very fast:(p

30 powm

References

The Internet Key Exchange (RFC 2409), Nov. 1998

Examples

packageDescription("gmp") # {possibly useful for debugging}

data(Oakley1)(M1

Random 31

Examples

powm(4,7,9)

x = as.bigz(4,9)x ^ 7

Random Generate a random number

Description

Generate a uniformly distributed random number in the range 0 to 2size − 1, inclusive.

Usage

urand.bigz(nb=1,size=200, seed = 0)

Arguments

nb Integer: number of random numbers to be generated (size of vector returned)size Integer: number will be generated in the range 0 to 2size − 1seed Bigz: random seed initialisation

Value

A biginteger of class bigz.

Author(s)

Antoine Lucas

References

‘mpz_urandomb’ from the GMP Library, see http://gmplib.org

Examples

# Integers are differentsurand.bigz()urand.bigz()urand.bigz()

# Integers are the sameurand.bigz(seed="234234234324323")urand.bigz(seed="234234234324323")

# Vectorurand.bigz(nb=50,size=30)

http://gmplib.org

32 sizeinbase

Relational Operator Relational Operators

Description

Binary operators which allow the comparison of values in atomic vectors.

Usage

## S3 method for class ’bigz’sign(x)## S3 method for class ’bigz’e1 == e2## S3 method for class ’bigz’e1 < e2

## S3 method for class ’bigz’e1 >= e2

Arguments

x, e1, e2 R object (vector or matrix-like) of class "bigz".

See Also

mod.bigz for arithmetic operators.

Examples

x

solve.bigz 33

Arguments

a big integer, i.e. "bigz"

b base

Value

integer of the same length as a: the size, i.e. number of digits, of each a[i].

Author(s)

Antoine Lucas

References


Examples

sizeinbase(342434, 10)# 6 obviously

Iv

34 Stirling

Details

It uses the Gauss and trucmuch algo . . . (to be detailled).

Value

If a and b are rational, return is a rational matrix.

If a and b are big integers (of class bigz) solution is in Z/nZ if there is a common modulus, of arational matrix if not.

Author(s)

Antoine Lucas

See Also

solve

Examples

x

Stirling 35

Usage

Stirling1(n, k)Stirling2(n, k, method = c("lookup.or.store", "direct"))Eulerian (n, k, method = c("lookup.or.store", "direct"))

Stirling1.all(n)Stirling2.all(n)Eulerian.all (n)

Arguments

n positive integer (0 is allowed for Eulerian()).

k integer in 0:n.

method for Eulerian() and Stirling2(), string specifying the method to be used."direct" uses the explicit formula (which may suffer from some cancelationfor “large” n).

Details

Eulerian numbers:A(n, k) = the number of permutations of 1,2,. . . ,n with exactly k ascents (or exactly k descents).

Stirling numbers of the first kind:s(n,k) = (-1)^n-k times the number of permutations of 1,2,. . . ,n with exactly k cycles.

Stirling numbers of the second kind:S(k)n is the number of ways of partitioning a set of n elements into k non-empty subsets.

Value

A(n, k), s(n, k) or S(n, k) = S(k)n , respectively.

Eulerian.all(n) is the same as sapply(0:(n-1), Eulerian, n=n) (for n > 0),Stirling1.all(n) is the same as sapply(1:n, Stirling1, n=n), andStirling2.all(n) is the same as sapply(1:n, Stirling2, n=n), but more efficient.

Note

For typical double precision arithmetic,Eulerian*(n, *) overflow (to Inf) for n ≥ 172,Stirling1*(n, *) overflow (to ±Inf) for n ≥ 171, andStirling2*(n, *) overflow (to Inf) for n ≥ 220.

Author(s)

Martin Maechler ("direct": May 1992)

36 Stirling

References

Eulerians:NIST Digital Library of Mathematical Functions, 26.14: http://dlmf.nist.gov/26.14

Stirling numbers:Abramowitz and Stegun 24,1,4 (p. 824-5 ; Table 24.4, p.835); Closed Form : p.824 "C."

NIST Digital Library of Mathematical Functions, 26.8: http://dlmf.nist.gov/26.8

See Also

chooseZ for the binomial coefficients.

Examples

Stirling1(7,2)Stirling2(7,3)

stopifnot(Stirling1.all(9) == c(40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1),Stirling2.all(9) == c(1, 255, 3025, 7770, 6951, 2646, 462, 36, 1),Eulerian.all(7) == c(1, 120, 1191, 2416, 1191, 120, 1)

)

http://dlmf.nist.gov/26.14http://dlmf.nist.gov/26.8

Index

!=.bigq (Bigq), 4!=.bigz (Relational Operator), 32∗Topic arithmetic

Stirling, 34∗Topic arith

apply, 2asNumeric, 3Bigq, 4bigq, 5Bigq operators, 7bigz, 8bigz operators, 11cumsum, 12Extract, 14Extremes, 15factorialZ, 16factorization, 17frexpZ, 19gcd.bigz, 20gcdex, 21gmp.utils, 22isprime, 23lucnum, 24matrix, 25modulus, 27nextprime, 28powm, 30Random, 31Relational Operator, 32sizeinbase, 32solve.bigz, 33

∗Topic characterformatN, 18

∗Topic dataOakley, 29

∗Topic mathis.whole, 22

∗Topic methodsasNumeric, 3

∗Topic printformatN, 18

*.bigq (Bigq operators), 7*.bigz (bigz operators), 11+.bigq (Bigq operators), 7+.bigz (bigz operators), 11-.bigq (Bigq operators), 7-.bigz (bigz operators), 11/.bigq (Bigq operators), 7/.bigz (bigz operators), 11=.bigz (Relational Operator), 32[.bigq (Extract), 14[.bigz (Extract), 14[

38 INDEX

array, 4as.bigq (bigq), 5as.bigz (bigz), 8as.bigz.bigq (bigq), 5as.character, 18as.character.bigq (bigq), 5as.character.bigz (bigz), 8as.double.bigq (bigq), 5as.double.bigz (bigz), 8as.matrix.bigq (matrix), 25as.matrix.bigz (matrix), 25as.numeric, 4as.vector.bigq (matrix), 25as.vector.bigz (matrix), 25asNumeric, 3asNumeric,ANY-method (asNumeric), 3asNumeric,bigq-method (asNumeric), 3asNumeric,bigz-method (asNumeric), 3asNumeric-methods (asNumeric), 3

biginteger_as (bigz), 8biginteger_as_character (bigz), 8Bigq, 4bigq, 5, 5, 7Bigq operators, 7bigz, 3, 7, 8, 11, 16, 19, 20, 26, 28, 29, 32, 33bigz operators, 11

c.bigq (Extract), 14c.bigz (Extract), 14cbind.bigq (matrix), 25cbind.bigz (matrix), 25character, 8chooseZ, 36chooseZ (factorialZ), 16class, 18crossprod (matrix), 25cumsum, 12, 12

denominator (bigq), 5denominator

INDEX 39

matrix, 4, 25, 27matrix.bigz, 3max, 15max.bigq (Extremes), 15max.bigz (Extremes), 15methods, 2, 15min, 15min.bigq (Extremes), 15min.bigz (Extremes), 15mod.bigz, 9, 32mod.bigz (bigz operators), 11modulus, 27modulus

package ‘gmp’package ‘gmp’ february 15, 2013 version 0.5-4 date 2013-02-04 title multiple...

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